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In mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used in differential topology. It is named after Henri Poincaré and Heinz Hopf. The Poincaré–Hopf theorem is often illustrated by the special case of the Hairy ball theorem, which simply states that there is no smooth vector field on a sphere having no sources or sinks. ==Formal statement== Let ''M'' be a differentiable manifold, of dimension ''n'', and ''v'' a vector field on ''M''. Suppose that ''x'' is an isolated zero of ''v'', and fix some local coordinates near ''x''. Pick a closed ball ''D'' centered at ''x'', so that ''x'' is the only zero of ''v'' in ''D''. Then we define the index of ''v'' at ''x'', indexx(''v''), to be the degree of the map ''u'':∂''D''→''S''n-1 from the boundary of ''D'' to the (''n''-1)-sphere given by ''u''(''z'')=''v''(''z'')/| ''v''(''z'') |. Theorem. Let ''M'' be a compact differentiable manifold. Let ''v'' be a vector field on ''M'' with isolated zeroes. If ''M'' has boundary, then we insist that ''v'' be pointing in the outward normal direction along the boundary. Then we have the formula : where the sum of the indices is over all the isolated zeroes of ''v'' and is the Euler characteristic of ''M''. The theorem was proven for two dimensions by Henri Poincaré and later generalized to higher dimensions by Heinz Hopf. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Poincaré–Hopf theorem」の詳細全文を読む スポンサード リンク
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